This linear algebra problem requires knowledge of the relationship between a matrix's eigenvalues, its determinant, and the determinants of its powers.
Step 1: Understanding the Question:
We are given the eigenvalues of a \(3 \times 3\) matrix \(A\) and are asked to find the determinant of the matrix \(A^2\), which is \(A\) multiplied by itself.
Step 2: Key Formula or Approach:
We can solve this in two ways, both relying on fundamental properties:
Method 1: Use the properties \( \det(A) = \prod \lambda_i \) and \( \det(A^k) = (\det A)^k \).
Method 2: First find the eigenvalues of \(A^2\) and then compute its determinant.
Step 3: Detailed Explanation (Method 1):
Find the determinant of A: The determinant of a matrix is the product of its eigenvalues.
\[ \det(A) = \lambda_1 \times \lambda_2 \times \lambda_3 = 1 \times 2 \times 3 = 6 \]
Find the determinant of A\(^2\): A property of determinants states that \( \det(A^k) = (\det(A))^k \). For \(k=2\):
\[ \det(A^2) = (\det(A))^2 = 6^2 = 36 \]
Step 3: Detailed Explanation (Method 2):
Find the eigenvalues of A\(^2\): If a matrix \(A\) has eigenvalues \( \lambda_i \), then the matrix \(A^k\) has eigenvalues \( \lambda_i^k \). Therefore, the eigenvalues of \(A^2\) are \(1^2, 2^2, 3^2\), which are \(1, 4, 9\).
Find the determinant of A\(^2\): The determinant of \(A^2\) is the product of its eigenvalues.
\[ \det(A^2) = 1 \times 4 \times 9 = 36 \]
Both methods yield the same result.
Step 4: Final Answer:
The determinant of \(A^2\) is 36.